Magnetic field from a conductor using Law of Biot-Savart

[Jon.G]

Homework Statement

An electric current I flows in a straight conductor of length L. Use the law of
Biot-Savart to find the magnetic field at a point lying on an axis going through the
centre of the conductor and perpendicular to the conductor.

Homework Equations

Law of Biot-Savart: $B=\frac{\mu _{0}}{4\pi }\int \frac{Idl \times \widehat{r}}{r^{2}}$

The Attempt at a Solution

Ok so this will be quite hard to explain my attempt so far without my diagram but here goes:
$r=\sqrt{x^{2} + y^{2}}$ where y is the height up the conductor (the 'position' of dl), x is the distance from the conductor along the x-axis

Let L go from -a to +a,
then $B=\frac{I \mu _{0}}{4\pi }\int^{+a}_{-a} \frac{dl \times \widehat{r}}{(x^{2} + y^{2})^{3/2}}$
which is the same as
$B=\frac{I \mu _{0}}{4\pi }\int^{2a}_{0} \frac{dl \times \widehat{r}}{(x^{2} + y^{2})^{3/2}}$

Then it's the whole dl x r bit that gets me. I'm sure I have to change this into dy, and then Iknow how to integrate that.

but I can't figure out how to bring dy into the equation :(

 

[George Jones]

Where is the origin of your $\left( x,y,z \right)$ coordinates?

 

[dauto]

Isn't dl = jdy?